Hadamard and Conference Matrices
K.T. Arasu
, Yu Qing Chen
and Alexander Pott
DOI: 10.1023/A:1011929727176
Abstract
We discuss new constructions of Hadamard and conference matrices using relative difference sets. We present the first example of a relative ( n, 2, n - 1, \frac n - 22) (n, 2, n - 1, \frac{{n - 2}}{2}) -difference set where n - 1 is not a prime power.
Pages: 103–117
Keywords: difference sets; relative difference sets; Hadamard matrices
Full Text: PDF
References
1. K.T. Arasu and S. Harris, “New constructions of group divisible designs,” J. Statist. Plann. Inference 52 (1996), 241-253.
2. A. Baliga and K.J. Horadam, “Cocyclic Hadamard matrices over Zt \times Z2, Australas. J. Comb. 11 (1995), 2 67-81.
3. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Vol. 1, 2nd edition, Cambridge University Press, Cambridge, 1999.
4. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Vol. 2, 2nd edition, Cambridge University Press, Cambridge, 1999.
5. C.J. Colbourn and J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996.
6. W. de Launey, D.L. Flannery, and K.J. Horadam, “Cocyclic Hadamard matrices and difference sets,” Discrete Appl. Math. 102 (2000), 47-61.
7. W. de Launey and K.J. Horadam, “A weak difference set construction for higher-dimensional designs,” Des. Codes Cryptogr. 3 (1993), 75-87.
8. W. de Launey and M.J. Smith, “Cocyclic orthogonal designs and the asymptotic existence of cocyclic Hadamard matrices and maximal size relative difference sets with forbidden subgroup of size 2,” J. Comb. Theory Ser. A 93 (2001), 37-92.
9. W. de Launey and R.M. Stafford, “On cocyclic weighing matrices and the regular group actions of certain Paley matrices,” Discrete Appl. Math. 102 (2000), 63-101.
10. P. Delsarte, J. Goethals, and J. Seidel, “Orthogonal matrices with zero diagonal. II,” Canad. J. Math. 23 (1971), 816-832.
11. S. Eliahou, M. Kervaire, and B. Saffari, “On Golay polynomial pairs,” Adv. Applied Math. 12 (1991), 235- 292.
12. D.L. Flannery, “Cocyclic Hadamard matrices and Hadamard groups are equivalent,” J. Algebra 192 (1997), 749-779.
13. J.C. Galati, “On the structure of groups containing central semiregular relative difference sets,” Research Report 1, Royal Melbourne Institute of Technology, 2001.
14. K.J. Horadam and W. de Launey, “Cocyclic development of designs,” J. Alg. Combin. 2 (1993), 267- 290.
15. N. Ito, “On Hadamard groups,” J. Algebra 168 (1994), 981-987.
16. N. Ito, “On Hadamard groups III,” Kyushu J. Math. 51 (1997), 369-379.
17. E. Johnsen, “Skew-Hadamard abelian group difference sets,” J. Algebra 4 (1966), 388-402.
18. D. Jungnickel, “On λ-ovals and difference sets,” in Contemporary Methods in Graph Theory, R. Bodendieck (Ed.), Bibliographisches Institut, Mannheim, 1990, pp. 429-448.
19. A. Pott, Finite Geometry and Character Theory, Vol. 1601 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 1995.
20. A. Pott, “A survey on relative difference sets,” in Groups, Difference Sets, and the Monster. Proceedings of a Special Research Quarter at the Ohio State University, Spring 1993, K.T. Arasu, J. Dillon, K. Harada, S. Sehgal, and R. Solomon (Eds.), Berlin, Walter de Gruyter 1996, pp. 195-232.
21. A. Pott, D. Reuschling, and B. Schmidt, “A multiplier theorem for projections of affine difference sets,” J. Stat. Plann. Inf. 62 (1997), 63-67.
22. B. Schmidt, “Williamson matrices and a conjecture of Ito's,” Des., Codes, Cryptogr. 17 (1999), 61-68.
23. J. Seberry and M. Yamada, “Hadamard matrices, sequences, and block designs,” in Contemporary Design Theory, Wiley, New York, 1992, pp. 431-560.
2. A. Baliga and K.J. Horadam, “Cocyclic Hadamard matrices over Zt \times Z2, Australas. J. Comb. 11 (1995), 2 67-81.
3. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Vol. 1, 2nd edition, Cambridge University Press, Cambridge, 1999.
4. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Vol. 2, 2nd edition, Cambridge University Press, Cambridge, 1999.
5. C.J. Colbourn and J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996.
6. W. de Launey, D.L. Flannery, and K.J. Horadam, “Cocyclic Hadamard matrices and difference sets,” Discrete Appl. Math. 102 (2000), 47-61.
7. W. de Launey and K.J. Horadam, “A weak difference set construction for higher-dimensional designs,” Des. Codes Cryptogr. 3 (1993), 75-87.
8. W. de Launey and M.J. Smith, “Cocyclic orthogonal designs and the asymptotic existence of cocyclic Hadamard matrices and maximal size relative difference sets with forbidden subgroup of size 2,” J. Comb. Theory Ser. A 93 (2001), 37-92.
9. W. de Launey and R.M. Stafford, “On cocyclic weighing matrices and the regular group actions of certain Paley matrices,” Discrete Appl. Math. 102 (2000), 63-101.
10. P. Delsarte, J. Goethals, and J. Seidel, “Orthogonal matrices with zero diagonal. II,” Canad. J. Math. 23 (1971), 816-832.
11. S. Eliahou, M. Kervaire, and B. Saffari, “On Golay polynomial pairs,” Adv. Applied Math. 12 (1991), 235- 292.
12. D.L. Flannery, “Cocyclic Hadamard matrices and Hadamard groups are equivalent,” J. Algebra 192 (1997), 749-779.
13. J.C. Galati, “On the structure of groups containing central semiregular relative difference sets,” Research Report 1, Royal Melbourne Institute of Technology, 2001.
14. K.J. Horadam and W. de Launey, “Cocyclic development of designs,” J. Alg. Combin. 2 (1993), 267- 290.
15. N. Ito, “On Hadamard groups,” J. Algebra 168 (1994), 981-987.
16. N. Ito, “On Hadamard groups III,” Kyushu J. Math. 51 (1997), 369-379.
17. E. Johnsen, “Skew-Hadamard abelian group difference sets,” J. Algebra 4 (1966), 388-402.
18. D. Jungnickel, “On λ-ovals and difference sets,” in Contemporary Methods in Graph Theory, R. Bodendieck (Ed.), Bibliographisches Institut, Mannheim, 1990, pp. 429-448.
19. A. Pott, Finite Geometry and Character Theory, Vol. 1601 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 1995.
20. A. Pott, “A survey on relative difference sets,” in Groups, Difference Sets, and the Monster. Proceedings of a Special Research Quarter at the Ohio State University, Spring 1993, K.T. Arasu, J. Dillon, K. Harada, S. Sehgal, and R. Solomon (Eds.), Berlin, Walter de Gruyter 1996, pp. 195-232.
21. A. Pott, D. Reuschling, and B. Schmidt, “A multiplier theorem for projections of affine difference sets,” J. Stat. Plann. Inf. 62 (1997), 63-67.
22. B. Schmidt, “Williamson matrices and a conjecture of Ito's,” Des., Codes, Cryptogr. 17 (1999), 61-68.
23. J. Seberry and M. Yamada, “Hadamard matrices, sequences, and block designs,” in Contemporary Design Theory, Wiley, New York, 1992, pp. 431-560.